J. Cent. South Univ. (2012) 19: 2496-2501
DOI: 10.1007/s11771-012-1302-0
Semi-active predictive control strategy for seismically excited structures using MRF-04K dampers
XU Long-he(徐龙河)1, LI Zhong-xian(李忠献)2
1. School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China;
2. School of Civil Engineering, Tianjin University, Tianjin 300072, China
? Central South University Press and Springer-Verlag Berlin Heidelberg 2012
Abstract: The theoretical study of a semi-active predictive control (SAPC) system with magnetorheological (MR) dampers to reduce the responses of seismically excited structures was presented. The SAPC scheme is based on a prediction model of the system response to obtain the control actions by minimizing an object function, which has a function of self-compensation for time delay occurring in real application. A double-ended shear mode combined with a valve mode MR damper, named MRF-04K damper, with the maximum force of 20 kN was designed and manufactured, and parameters of the Bouc-Wen hysteresis model were determined to portray the behavior of this damper. As an example, a 5-story building frame equipped with 2 MRF-04K dampers was presented to demonstrate the performance of the proposed SAPC scheme for addressing time delay and reducing the structural responses under different earthquakes. Comparison with the uncontrolled structure, the passive-off and passive-on cases indicates that both the peak and the norm values of structural responses are all clearly reduced, and the SAPC scheme has a better performance than the two passive cases.
Key words: semi-active control; predictive control; time delay compensation; magnetorheological (MR) damper
1 Introduction
The field of structural control is becoming increasingly important in civil engineering as taller and more flexible structures are built, which are more vulnerable to strong wind and earthquakes. Among these control methods, semi-active control technique has received much attention and been demonstrated to have a great deal of promise for civil engineering applications. Especially, the occurrence of some smart materials and controllable dampers, such as electrorheological (ER) dampers and magnetorheological (MR) dampers, makes the semi-active control technique more practical and feasible. MR dampers are quite promising for civil engineering applications, since they have many attractive features such as small power requirements, reliability, and being inexpensive to manufacture. Many analytical and experimental studies have been performed on their behavior and application to civil structures, and have shown better control performance in reducing seismic responses [1-8].
Various control strategies have been evaluated and compared for semi-active control systems with MR dampers in numerical studies [9]. In this work, we focus on another control strategy, semi-active predictive control (SAPC) scheme, which is based on a prediction model of the system response to obtain the control actions by minimizing an object function. The prescribed optimization objective is determined by minimizing the difference between the predicted and target responses. For the entire control progress, time delay is one problem which needs a serious attention, and various methodologies to deal with it have been proposed [10-18]. And the SAPC scheme has a function of self-compensation for time delay that occurs in real application.
A kind of MR damper, named the MRF-04K damper, had been developed and manufactured, and its dynamic performance had been experimentally studied [6]. The maximum force at full magnetic field intensity is about 20 kN while the maximum power required is less than 50 W. As a numerical example, a 5-story frame structure equipped with 2 MRF-04K dampers is analyzed to demonstrate the validity of the SAPC scheme for addressing time delay and reducing the structural responses under different earthquakes.
2 Predictive control
For an n-degree of freedom building with r control devices subjected to seismic excitation üg(t), the discrete equation of motion is written as
(1)
where z(k) is a 2n-dimensional state vector, 
is a r-dimensional control action vector, in which 
(
) is the time delay and ?t is the sampling period,
is the external excitation, G is a 2n×2n state matrix, H is a 2n×r matrix, and W1 is a 2n-dimensional vector.
Referring to Eq. (1), the predictive model is defined as
(2)
where 
is a 2n-dimensional state vector at a future sampling period, k+j, estimated by the information available at time step k, 
is a r-dimensional predictive control vector, and l is predictive length. This model can be auto-updated at the k-th time step, that is,
;
(3)
Within the time step from k to k+l, the objective function Jk can be defined as
(4)
where Qj and Rj are 2n×2n and r×r weighting matrices, respectively. Substituting Eq. (3) into Eq. (2), the predicted state vector at the subsequent time steps k+j, j=1, 2, …, l+d, can be expressed as a function of the current state vector z(k) and the control force vector u(k), and the last predictive equation can be written as
(5)
Define dr-dimensional and lr-dimensional control force vectors as
(6)
The (d+l)2n-dimentional predictive state vector can be written as
(7)
where G1, H1 and H2 are (d+l)2n×2n, (d+l)2n×dr, and (d+l)2n×lr state matrices, respectively.
Equation (4) can be rewritten as
(8)
where Q and R are (d+l)2n×(d+l)2n and lr×lr weighting matrices, respectively.
Differentiating Jk with respect to U, 
, the optimal predictive control force is given by
(9)
where
(10)
At each time step, the control force u(k) is taken as
(11)
where D1 is the first r row of matrix LG1, and D2 is the first r row of matrix LH1.
In order to simplify the computation of Eq. (11) for higher order system, assume that control action is uniform over the duration of predictive period of time, that is,
(12)
Therefore, the following objective function is selected:
(13)
where Q′ and R′ are 2n×2n and r×r weighting matrices, respectively.
Substituting Eq. (12) into Eq. (5), there is
(14)
where
(15)
By differentiating Jk with respect to
, the optimal predictive control force is given by
(16)
where
(17)
The prediction of future multi-step state 
is based both on actual past control action u(k), u(k-1), …, u(k-d) and on actual current measuring response state z(k) to reach a prescribed optimization objective, minimizing the difference between the predicted and target responses. The control command determined by the prediction model is then applied to the structure, the actual state of the next step will be measured, and the results of comparison with the predicted one are utilized to update future predictions.
3 MRF-04K damper
A double-ended, shear mode combined with valve mode MRF-04K damper has been designed and manufactured. Figure 1 shows the picture of this damper, which has an inside diameter of 12.5 cm and a stroke of ±4 cm, approximately 0.5 m long and with a mass of 50 kg. The maximum force at a full magnetic field intensity is 21.25 kN at a piston velocity of 12.57 cm/s [6].
Fig. 1 MRF-04K damper
The MRF-04K damper is used as semi-active control device and employed in the SAPC system, and a simple Bouc-Wen model [7] is used to portray the behavior of this damper. The equation governing the force produced by the damper is
(18)
And the Bouc-Wen element is
(19)
where x and F(t) are the displacement and force of the damper, respectively, x0 is the initial displacement of spring k0, and z(t) is the evolutionary variable that accounts for the hysteretic behavior of the device. The parameters c0 and α are future functions of applied voltage u:
(20)
The parameters of Bouc-Wen model are selected as follows: c0a=80 N·s/cm, c0b=15 N·s/cm·V, k0=10 N/cm, x0=18.6 cm, aa=2.1 kN/cm, ab=1.7 kN/cm·V, g=30 cm-2, b=30 cm-2, n=2, and A=60. The typical predicted and experimental responses of the MRF-04K damper due to 1.0 Hz sinusoidal excitation with amplitude of 10 mm are shown in Fig. 2 for five voltage levels of 0, 2.5, 5.0, 7.5, and 10 V, respectively. The Bouc-Wen model can accurately portray the behavior of the MRF-04K damper.
Fig. 2 Experimental and predicted damper force due to 1.0 Hz sinusoidal excitation with amplitude of 10 mm: (a) Force versus time; (b) Force versus displacement; (c) Force versus velocity
In real application, the control force F(t) produced by the MRF-04K damper can be controlled by adjusting the voltage applied to the current driver connected with the damper, so the simple bang-bang control law is defined as
(21)
where Vmax is the maximum applied voltage. If the magnitude of the force Fi(t) produced by the i-th device is smaller than that of the desired optimal force ui(t) and the two forces have the same sign, the voltage applied to the i-th damper is increased to the maximum level; otherwise, the commanded voltage is set to be zero.
4 Numerical example
A 5-story frame structure equipped with two MRF-04K dampers is examined, and the structural properties are given in Table 1. Two historical earthquake records, El Centro (1940 NS) and Northridge (1994 NS), are considered, and the peak acceleration of the ground motion is taken as 0.2g. Two MRF-04K dampers are installed in combination with steel braces on the first floor and the third floor, assuming that the steel braces are infinitely rigid.
Table 1 Properties of controlled structure
The control performance of the SAPC system with MR dampers declines while the predictive length (l) exceeds a specific value [18]. In the numerical analysis, the selection of predictive length l=10 and the delayed time step d=10 is considered. For El Centro and Northridge earthquakes, the sampling periods are 0.02 and 0.01 s, respectively, and the time delay magnitudes are 0.2 and 0.1 s, respectively.
The performance of the SAPC strategy is evaluated with 10 criteria (J1-J6 and J11-J14) provided by ASCE benchmark control problems [19]. J1-J6 are related to the building responses, and J11-J14 are related to the control devices. A comparison between the SAPC strategy with l=10 and d=10 and passive-off (0 V) and passive-on (10 V) methods is shown in Table 2. It can be observed that the norm values (J4, J5, and J6) are all reduced more with the SAPC strategy than those with two passive methods, and the peak values (J1, J2, and J3) are all reduced more than those with passive-off method under various earthquakes. These results reflect that the SAPC system with MRF-04K dampers are more effective than the passive methods in reducing structural responses subjected to earthquakes.
The representative responses at the top floor of the SAPC system with time delay compensation for the two scaled earthquake records are shown in Figs. 3 and 4, which are compared with those of the uncontrolled structure. Control force and required voltage time histories of MRF-04K damper on the first floor are shown in Figs. 5 and 6. Both the peak drift ratio and level acceleration are all reduced. For the SAPC system with MRF-04K dampers, the peak drift ratio (J1) and the peak level acceleration (J2) of the 5-story building are reduced by 18% and 11%, respectively, due to El Centro earthquake, and 21% and 20%, respectively, due to Northridge earthquake. And the peak base shear is reduced by 8% and 6%, respectively, due to two earthquakes. However, the norm values (J4, J5, and J6) are all reduced significantly for the two earthquakes. The maximum control force produced by MRF-04K damper on the first floor is 8.3 and14.3 kN, respectively, over the duration of El Centro and Northridge earthquakes.
Table 2 Control performance of SAPC strategy versus passive method
Fig. 3 Controlled and uncontrolled roof responses for El Centro earthquake: (a) Displacement; (b) Acceleration
Fig. 4 Controlled and uncontrolled roof responses for Northridge earthquake: (a) Displacement; (b) Acceleration
Fig. 5 Control force (a) and applied voltage (b) vs time histories of MRF-04K damper on first story during El Centro earthquake
Fig. 6 Control force (a) and applied voltage (b) vs time histories of MRF-04K damper on first story during Northridge earthquake
5 Conclusions
1) A semi-active predictive control (SAPC) system using MR dampers is proposed to reduce the seismic responses of structures, which is based on a prediction model of the system response to obtain the control actions by minimizing an object function and has a function of self-compensation for time delay that occurs in real application.
2) As a numerical example, a 5-story building equipped with two 20 kN MRF-04K dampers is introduced to demonstrate the validity of this approach. Two historical earthquake records, El Centro (1940 NS) and Northridge (1994 NS), all scaled to the peak acceleration of 0.2g are used as the input ground excitations.
3) Comparison with the uncontrolled structure indicates that both the peak values and the norm values of structural responses are all clearly reduced when MRF-04K dampers are used in the SAPC system and the predictive length l=10 and the delayed time step d=10 are selected.
4) Comparison with two passive control methods is also made. The results indicate that the SAPC system with MRF-04K dampers are more effective than the passive methods in reducing structural responses subjected to earthquakes.
References
[1] Jr SPENCER B F, CARLSON J D, SAIN M K, YANG G. On the current status of magnetorheological dampers: Seismic protection of full-scale structures [C]// Proceedings of American Control Conference. Albuquerque, New Mexico, 1997: 458-462.
[2] DYKE S J, Jr SPENCER B F, SAIN M K, CARLSON J D. Seismic response reduction using magnetorheological dampers [C]// Proceedings of IFAC World Congress, San Francisco, CA, 1996.
[3] CARLSON J D, Jr SPENCER B F. Magneto-rheological fluid dampers for semi-active seismic control [C]// Proceedings of 3rd International Conference on Motion and Vibration Control. Chiba, Japan, 1996: 35-40.
[4] CARLSON J D, Jr SPENCER B F. Magnetorheological fluid dampers: Scalability and design issues for application to dynamic hazard mitigation [C]// Proceedings of 2nd International Workshop on Structural Control. Hong Kong, 1996: 99-109.
[5] HIEMENZ G J, CHOI Y T, WERELEY N M. Seismic control of civil structures utilizing semi-active MR braces [J]. Comput-aided CIV INF, 2003, 18(1): 31-44.
[6] LI Zhong-xian, XU Long-he. Performance tests and hysteresis model of MRF-04K damper [J]. Journal of Structural Engineering, ASCE, 2005, 131(8): 1303-1306.
[7] Jr SPENCER B F, DYKE S J, SAIN M K, CARLSON J D. Phenomenological model for magnetorheological dampers [J]. Journal of Structural Engineering, ASCE, 1997, 123(3): 230-238.
[8] YANG G, Jr SPENCER B F, JUNG H J, CARLSON J D. Dynamic modeling of large-scale magnetorheological damper systems for civil engineering applications [J]. Journal of Engineering Mechanics, ASCE, 2004, 139(9): 1107-1114.
[9] JANSEN L M, DYKE S J. Semi-active control strategies for MR dampers: Comparative study [J]. Journal of Engineering Mechanics, ASCE, 2000, 126(8): 795-803.
[10] YANG J N, AKBARPOUR A, ASKAR G. Effect of time delay on control of seismic-excited buildings [J]. Journal of Structural Engineering, ASCE, 1990, 116(10): 2801-2814.
[11] AGRAWAL A K, YANG J N. Compensation of time-delay for control of civil engineering structures [J]. Earthquake Engineering and Structural Dynamics, 2000, 29: 37-62.
[12] CHU S Y, SOONG T T, LIN C C, CHEN Y Z. Time-delay effect and compensation on direct output feedback controlled mass damper systems [J]. Earthquake Engineering and Structural Dynamics, 2002, 31(1): 121-137.
[13] AGRAWAL A K, YANG J N. Effect of fixed time-delay on stability and performance of actively controlled civil engineering structures [J]. Earthquake Engineering and Structural Dynamics, 1997, 26(11): 1169-1185.
[14] INAUDI J A, KELLY J M. A robust delay-compensation technique based on memory [J]. Earthquake Engineering and Structural Dynamics, 1994, 23(9): 987-1001.
[15] CAI G P, HUANG J Z. Optimal control method for seismically excited building structures with time-delay in control [J]. Journal of Engineering Mechanics, ASCE, 2002, 128(6): 602-612.
[16] KEVIN K F WONG. Predictive optimal linear control of elastic structures during earthquake. Part I [J]. Journal of Engineering Mechanics, ASCE, 2005, 131(2): 131-141.
[17] KEVIN K F WONG. Predictive optimal linear control of inelastic structures during earthquake. Part II [J]. Journal of Engineering Mechanics, ASCE, 2005, 131(2): 142-152.
[18] XU Long-he, LI Zhong-xian. Semi-active multi-step predictive control of structures using MR dampers [J]. Earthquake Engineering and Structural Dynamics, 2008, 37: 1435-1448.
[19] OHTORI Y, CHRISTENSON R E, SPENCER Jr B F, DYKE S J. Benchmark control problems for seismically excited nonlinear buildings [J]. Journal of Engineering Mechanics, ASCE, 2004, 130(4): 366-385.
(Edited by YANG Bing)
Foundation item: Projects(90815025, 51178034) supported by the National Natural Science Foundation of China
Received date: 2011-09-09; Accepted date: 2011-12-28
Corresponding author: XU Long-he, Associate Professor, PhD; Tel: +86-10-51684953; E-mail: lhxu@bjtu.edu.cn
